We investigate the interplay between adsorption and transport in a two-dimensional porous medium by means of an extended Lattice Boltzmann technique within the Two-Relaxation-Time framework. We focus on two canonical adsorption thermodynamics and kinetics formalisms: (1) the Henry model in which the adsorbed amount scales linearly with the free adsorbate concentration and (2) the Langmuir model that accounts for surface saturation upon adsorption. We simulate transport of adsorbing and nonadsorbing particles to investigate the effect of the adsorption/desorption ratio k, initial free adsorbate concentration c0, surface saturation Γ∞, and Peclet numbers Pe on their dispersion behavior. In all cases, despite marked differences between the different adsorption models, the three following transport regimes are observed: diffusion-dominated regime, transient regime and Gaussian or nearly Gaussian dispersion regime. On the one hand, at short times, the intermediate transient regime strongly depends on the system's parameters with the shape of the concentration field at a given time being dependent on the amount of particles adsorbed shortly after injection. On the other hand, at longer times, the influence of the initial condition attenuates as particles sample sufficiently the adsorbed and nonadsorbed states. Once such dynamical equilibrium is reached, transport becomes Gaussian (i.e., normal) or nearly Gaussian in the asymptotic regime. Interestingly, the characteristic time scale to reach equilibrium, which varies drastically with the system's parameters, can be much longer than the actual simulation time. In practice, such results reflect many experimental situations such as in water treatment where dispersion is found to remain anomalous (non-Gaussian), even if transport is considered over long macroscopic times.